Oscillator synchronization in complex networks with non-uniform time delays

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1 Oscillator synchronization in complex networks with non-uniform time delays Jens Wilting 12 and Tim S. Evans 13 1 Networks and Complexity Programme, Imperial College London, London SW7 2AZ, United Kingdom 2 Dept. of Epileptolgy, University of Bonn, Sigmund-Freud-Strasse 25, D-5315 Bonn, Germany jwilting@uni-bonn.de 3 Theoretical Physics, Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom t.evans@imperial.ac.uk Abstract. We investigate a population of limit-cycle Kuramoto oscillators coupled in a complex network topology with coupling delays introduced by finite signal propagation speed and embedding in a ring. By numerical simulation we find that in complete graphs velocity waves arise that were not observed before and analytically not understood. In regular rings and small-world networks frequency synchronization occurs with a large variety of phase patterns. While all these patterns are nearly equally probable in regular rings, small-world topology sometimes prefers one pattern to form for a large number of initial conditions. We propose implications of this in the context of the temporal coding hypothesis for information processing in the brain and suggest future analysis to conclude the work presented here. Introduction The study of coupled oscillators and their synchronization behavior [3] has been a field of study for a long time with widely spread applications like flashing fireflies, cardiac pacemaker cells or epileptic seizures [15,17]. Furthermore the interplay of oscillators is also subject of interest in the context of the temporal coding hypothesis [5,7] which suggests that the information emitted by a population of neurons is encoded by the timing of firing action potentials, and therefore depends on the coupling within this population. A simple model for synchronization of limit-cycle phase oscillators is given by the Kuramoto model [1,14]. If the coupling is constraint by an underlying network topology, i.e. two oscillators or nodes are coupled if there is a link between them in the network, it has been shown that the network structure itself influences the synchronization of oscillators [2,4,8,9,1,16,19] and can either favor or hinder coherence. In most applications the coupling cannot be regarded as instantaneous but finite signal propagation speeds impose delays, e.g. the speed of sound or axonal signal speeds in nerve cells. In the case of identical delays for each pair of coupled oscillators this leads to vastly enhanced synchronization phenomenology [6,22].

2 However often one would not expect the delays to be identical for all oscillators. Then in the most general form the Kuramoto model for N limit-cycle phase oscillators is extended to the form θ i (t) = ω i 1 k i N j=1k i j sin(θ i (t) θ j (t τ i j )). (1) with the degree k i of node i to incorporate non-uniform time delays τ i j and coupling strengths K i j between oscillators i and j with phases θ i and θ j respectively. The underlying network topology with adjacency matrix A i j is encoded in the coupling strengths such that K i j = K A i j. In the case of delays induced by signal propagation over a distance d i j with finite signal speed v the delays are given as τ i j = d i j /v. Since networks do not have an inherent metric the oscillator population is embedded on a one-dimensional unit circle [23] with oscillator positions φ i. The distance between two oscillators is then given by the minimum distance on this circle, d i j = min { φ i φ j, φ i φ j }, (2) which behaves like a well-defined metric. The oscillators are supposed to be evenly spread around the ring, thus the positions are given by φ i = N (i 1). (3) In this setup with global coupling K i j = K for all i, j depending on the propagation speed full synchronization and waves traveling around the ring arise [23], a feature that was also observed in a two-dimensional metric space [11] and if random graphs are chosen for the coupling topology [12,13]. Since analytical results can only be obtained for a limited number of cases, one of them being presented for global coupling in [23], we make wide use of numerical simulations. A fourth order Runge Kutta scheme, extended by a backwards search in time to incorporate the time delays, is used for integration. Since the delays require an evaluation at t < the system is integrated from t = to t = max { τ i j } with a backwards Euler method and free oscillator evolution θ = ω i, i.e. the coupling is supposed to set in at t = and is in fact given by K i j H(t) with the Heavyside step function H(t). In accordance with previous work on this particular model we choose identical frequencies = ω i = 1. and K = 1. while the signal speed is drawn from the interval v [.15,1.35]. The system is simulated from t = max { τ i j } to t = 2 with a step size of.25. Every simulation is repeated twenty times with different sets of initial conditions θ i () drawn from a uniform random distribution in [,) to reveal potentially critical dependence on initial conditions. Different types of networks with 1 nodes each are chosen for the underlying coupling topology. Global coupling With all-to-all coupling, K i j = K for all i, j, we reproduce the results of [23] and find that for all initial conditions the system converges to full phase coherence or phase

3 Fig. 1. Final state winding number observed for at least one set of initial conditions versus signal propagation 5 4 speed. m (v) is uniquely 3 defined except near the transitions to higher m, where different initial conditions may lead to different winding numbers. 2 1 Observed m /v waves traveling around the ring, whose winding number m depends on the size of the delays. It can be extracted from the simulation by summation over all phase differences between pairs of adjacent oscillators, where the ambiguity of shifts is resolved by requiring the difference to be within [,]: m(t) = 1 N i=1 x, if x f (θ i (t) θ i+1 (t)), where f (x) = x, if x > x +, if x <. Analysis reveals that the winding number defined in this way after initial fluctuations converges to a stable value. These fluctuations represent the transient from initial conditions to the observed waves: From randomly distributed initial conditions, yielding random winding number m, growing segments of the ring synchronize or form waves. These waves may have different directions so that typically the winding number goes down to a value close to at some point, until two large segments with opposite waves form. When these segments gradually merge into one globally coherent pattern the winding number converges to its final value. Figure 1 shows that the final winding number observed in the system increases with the delays, i.e. with 1/v. This increase appears to be nearly linear in the investigated parameter range, but is quantized to integer values of the winding number. Furthermore the mapping of 1/v to m is uniquely defined except near the transitions to higher m where two different winding numbers can manifest dependent on the initial conditions. The one exception at 1/v 6.7 is due to the fact that for large delays the transient to a stable wave-mode is not completed when the simulation finishes at t = 2, so that at this time two large segments with opposite waves exist, canceling their phase differences to m =. Zanette [23] observed that all oscillators, whose velocity is Ω i = θ i, lock to one common frequency Ω so that the velocity dispersion σ = (Ω i Ω ) 2 decays with a decay constant dependent on the size of the delays. We confirm that for high signal speed, i.e. small delays, this is indeed the case but for signal speed v <.75 find that the velocity dispersion converges to a finite limit (see Fig. 2). Closer analysis of the velocities shows that for each oscillator Ω i is not a constant value but periodically (4)

4 Fig. 2. The velocity dispersion σ decays exponentially for v >.95 but converges to a finite limit for smaller speeds. This is the result of each single oscillator velocity oscillating around an average value, so that in fact the velocity forms a wave traveling around the ring in addition to the known phase waves v=1.15 v=1.35 v=.95 v= t oscillates around Ω. The phase of these oscillations is shifted by for opposite oscillators, so that in fact also the instantaneous velocity of the oscillators forms a wave traveling around the ring. The peak-to-peak amplitude of this oscillation ranges from.5 up to.8 compared to an average velocity of Ω.9. So far we do not know why these velocity oscillations arise and why they were not seen before. We observe them consistently for all initial conditions and with a simulation step size varied over 2.5 decades, making it seem improbable that they are a mere artifact of the simulation. Furthermore the limit of the velocity dispersion depends only on v and is the same for all initial conditions if v is fixed. Attempts to understand these waves as a perturbations of the frequency-locked wave modes that were analytically understood [23] have not been successful so far and are restricted to assumptions that are valid in a very limited number of observed cases only, so that more effort will be needed to explain the observed phenomenon. Regular rings and small-worlds Regular rings are inherently connected to the embedding in a circle. Therefore we examine networks in which each node is connected to its k nearest neighbors where k is even. We observe that in all cases the oscillators synchronize in frequency after an initial transient and converge to a stable state where they lock to one phase pattern and propagate at a common frequency. However this phase pattern turns out to be different for every realization of initial conditions in a bidirectional ring with only nearest neighbor connections and average degree k = 2. Furthermore more complex patterns than waves occur as is depicted in the first row of Fig. 3. For an explanation of this difference from the globally connected case follow the analysis in [23] and assume that the oscillators lock to a common frequency Ω and

5 /2 3/2 5 / /2 3/2 5 / /2 5 /2 3/2 5 1 / /2 5 / Fig. 3. Spatio-temporal phase representation for regular rings and v =.35. In all cases a steady state evolution with a stable propagating phase pattern emerges. First row: In a bidirectional ring with k = 2 a different, possibly complicated, phase pattern emerges for each set of initial conditions. Third row: Only full synchronization and wave modes occur for k = 1. evolve as θi (t) = Ωt + θ i. Substituting this in the model equation (1) yields Ω =ω 1 ki N Ki j sin Ω τi j + θ i θ j. (5) j=1 For global coupling Ki j = K the requirement of the left hand side of this equation being independent of i restricts possible choices of the θ i and leads to wave modes [23]. In a ring however only two terms contribute to the sum and give the new condition Ω =ω K sin Ω τ + θ i θ i 1 + sin Ω τ + θ i θ i+1 2 (6) where τ = τi,i±1. Define i = θ i θ i 1, then then a solution consistent with frequency synchronization must give the same result Ω for all i, i.e. sin (Ω τ + i ) + sin (Ω τ i+1 ) = const. (7) By fixing two phases θ i, θ i+1 this equation fixes all other θ j. Note however that only solutions for i are possible that sum up to integer multiples of and must comply with the self-consistency equation (6) for Ω. Nevertheless this equation imposes less constraints on possible phase differences between oscillators, so that more phase patterns are consistent with locking to a common frequency Ω and can arise. This conforms to the observation that in regular rings with coupling to the next five neighbors on each side, where more terms contribute to the sum in equation (5) and impose more constraints, only wave modes and complete synchronization are observed (second row in Fig. 3) similar to the case of global coupling in [23]. From regular rings small-world networks are created [2] by rewiring links to a random new destination with probability p and thus adding shortcuts and disorder into the network. We inspect networks with average degrees 4 and 12 and find that the system

6 behavior strongly depends on the rewiring probability p: In the limit of regular rings, p <.5, the behavior of rings is reproduced, i.e. wave modes manifest for higher degree k = 12 while a large variety of stable phase configurations is observed for k = 4. In contrast to the purely regular case there are phase slips that divide the pattern into separate coherent parts. These slips can be mapped to the few shortcuts in the system. Increasing the amount of disorder in the network for p >.5 the observations for random graphs [12,13] are reproduced: The stable phase patterns show no obvious coherence for k = 4 and are waves for k = 12. More connected graphs show an interesting behavior: While in both limits of p waves are observed, the range of intermediate.1 < p <.5 produces a large variety of phase patterns as seen in regular rings. However there is an important difference, as some network realizations seem to highly favor one certain pattern to form for a large number of initial conditions. Fig. 4. Defining an oscillator to fire when passing a phase of from below a firing sequence similar to those in [21] is observed in one special realization of small-world topology with p =.3 for 8% of initial conditions. Other smallworld realizations may lead to other firing patterns with large basin of attraction or have no pattern produced by a significantly large percentage of initial conditions In the light of neuronal networks and the temporal coding hypothesis [5] it has been shown [18,21] that rings of unidirectionally coupled oscillators can produce predefined temporal firing sequences by adjusting the coupling delays appropriately as can be done by synaptic plasticity in the brain. The results obtained in this work suggest that also special network topology can be used to generate one certain firing pattern with high probability. Defining an oscillator to fire when passing a phase of from below, Fig. 4 depicts the periodically occurring firing sequence of an oscillator population that was observed for 8% of initial conditions. A large variety of other patterns with large basin of attraction was observed for other realizations of small-world networks. So far we have not fully understood the connection between network and resulting pattern, although there are hints for a connection between high closeness centrality of a section of the ring and late firing in the sequence of the corresponding oscillators and early firing for oscillator sections with lower closeness centrality which would indicate a connection to information spread within the network.

7 Conclusion In summary we have investigated the synchronization of limit-cycle oscillators with distance dependent coupling delays introduced by finite signal propagation speed. For global coupling topology we mostly confirm prior results [13,23] but find differences in the frequency synchronization, where we observe frequency waves traveling around the ring in addition to the phase waves. The smaller number of coupling partners in (7) for regular rings allows more phase patterns to stably exist in the oscillator population consistent with frequency locking. While for regular rings the developing pattern was different for almost every set of initial conditions, some realizations of small-world connectivity with a small amount of disorder or shortcuts result in a large basin of attraction of one certain phase pattern, which is favored to form for a large number of initial conditions. This has implications for the creation of temporal firing patterns which seems possible through special smallworld topology. From this point many directions are open for future research like breaking the symmetry of evenly spread oscillator positions on the ring or inhomogeneous coupling constants. Acknowledgments We would like to thank Michael Gastner, Tiago Pereira, Jeldtoft Jensen and Gunnar Pruessner for their advice and help. References 1. Acebrón, J.A., Bonilla, L.L., Pérez Vicente, C. J. et al.: The Kuramoto model: A simple paradigm for synchronization phenomena. Rev. Mod. Phys. (25) doi:1.113/revmodphys Arenas, A., Díaz-Guilera, A., Kurths, J. et al: Synchronization in complex networks. Phys. Rep. (28) doi:1.116/j.physrep Boccaletti, S., Kurths, J., Osipov, G. et al.: The synchronization of chaotic systems. Phys. Rep. (22) doi:1.116/s (2) Brede, M.: Locals vs. global synchronization in networks of non-identical Kuramoto oscillators. Eur. Phys. J. B (28) doi:1.114/epjb/e Cessac, B., Paugam-Moisy, H., Viéville, T.: J. Physiol. Paris (21), doi:1.116/j.jphysparis Choi, M.Y., Kim, H.J., Kim, D., Hong, H.: Synchronization in a system of globally coupled oscillators with time delay. Phys. Rev. E (2) doi:1.113/physreve Foss, J., Longtin, A., Mensour, B., Milton, J.: Multistability and delayed recurrent loops. Phys. Rev. Lett. (1996) doi:1.113/physrevlett Gomez-Gardenes, J., Moreno, Y., Arenas, A.: Paths to Synchronization on Complex Networks. Phys. Rev. Lett. (27) doi:1.113/physrevlett Hong, H. and Choi, M.Y. and Kim, B.J.: Synchronization on small-world networks. Phys. Rev. E (22) doi:1.113/physreve Ichinomiya, T.: Frequency synchronization in a random oscillator network. Phys. Rev. E (24) doi:1.113/physreve Jeong, S.-O., Ko, T.-W., Moon, H.-T.: -Delayed Spatial Patterns in a Two-Dimensional Array of Coupled Oscillators. Phys. Rev. Lett. (22) doi:1.113/physrevlett

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